Problem: $ -2.\overline{4} \div 0.\overline{59} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 10x &= -24.4445...\\ x &= -2.4445...\end{align*} $ $\begin{align*} 9x &= -22 \\ x &= -\dfrac{22}{9}\end{align*} $ $\begin{align*} 100y &= 59.5959...\\ y &= 0.5959...\end{align*} $ $\begin{align*} 99y &= 59 \\ y &= \dfrac{59}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{22}{9} \div \dfrac{59}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{22}{9} \times \dfrac{99}{59} = {?} $ $ \phantom{-\dfrac{22}{9} \times \dfrac{59}{99}} = \dfrac{-22 \times 99}{9 \times 59} $ $ \phantom{-\dfrac{22}{9} \times \dfrac{59}{99}} = \dfrac{-22 \times \cancel{99}11} {\cancel{9} \times 59} $ $ \phantom{-\dfrac{22}{9} \times \dfrac{59}{99}} = -\dfrac{242}{59} $